The equation of the circle circumscribing the triangle formed by the straight lines $x+y=6$,$2x+y=4$,and $x+2y=5$ is given by:

Suppose $Q$ is a point on the circle with centre $P$ and radius $1$,as shown in the figure. $R$ is a point outside the circle such that $QR = 1$ and $\angle QRP = 2^{\circ}$. Let $S$ be the point where the segment $RP$ intersects the given circle. Then,the measure of $\angle RQS$ equals $......^{\circ}$.

Let $A, B, C$ be three points on a circle of radius $1$ such that $\angle ACB = \frac{\pi}{4}$. Then,the length of the side $AB$ is

Choose the correct statement about two circles whose equations are given below:
$x^{2}+y^{2}-10x-10y+41=0$
$x^{2}+y^{2}-22x-10y+137=0$

The number of common tangents to the circles $x^2 + y^2 = 1$ and $x^2 + y^2 - 4x + 3 = 0$ is

If $A(1,2)$ and $B(2,1)$ are two vertices of an acute-angled triangle and $S(0,0)$ is its circumcentre,then the angle subtended by $AB$ at the third vertex is